Memory flows: geometry and dynamics of sequential retrieval in input-driven Hopfield networks
Abstract
Associative memory models classically describe retrieval as convergence to stored patterns, but the theory of sequential retrieval remains comparatively limited. We develop a two-timescale input-driven Hopfield network in which fast associative retrieval is modulated by a slow feedback variable evolving over a prescribed memory-transition graph. This yields autonomous, graph-constrained transitions while preserving analytical tractability. Using slow-fast and geometric singular perturbation theory, we characterize memory fixation via memory manifolds and identify the geometric regions that govern stability loss and escape from retrieved memories. We further derive reduced slow dynamics and explicit criteria for transition onset, self-sustained retrieval, and collapse, expressed through escape times, gain thresholds, and fixed points. Together, these results provide a tractable framework for analyzing feedback-driven sequential retrieval in Hopfield networks.
Related
- Neuromodulation-inspired gated associative memory networks: Extended memory retrieval and emergent multistability
- RA-PLA: Retrieval-augmented graph convolutional networks for protein-ligand binding affinity prediction
- Sequential syntax of EEG microstates captures the temporal dynamics of inhibitory control
- Sequential chaotic oscillations in excitatory-inhibitory threshold-linear networks
- Estimating latent neuronal nonlinear dynamics by sequential Monte Carlo method and sparse modeling
BibTeX
@misc{betteti2026memory,
title = {{Memory flows: geometry and dynamics of sequential retrieval in input-driven Hopfield networks}},
author = {Simone Betteti and Matteo Martin and Morten Gram Pedersen and Sandro Zampieri and Giacomo Baggio},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09995},
eprint = {2611.09995},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09995},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Memory flows: geometry and dynamics of sequential retrieval in input-driven Hopfield networks
AU - Simone Betteti
AU - Matteo Martin
AU - Morten Gram Pedersen
AU - Sandro Zampieri
AU - Giacomo Baggio
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.09995
UR - https://arxiv.org/abs/2611.09995
AB - Associative memory models classically describe retrieval as convergence to stored patterns, but the theory of sequential retrieval remains comparatively limited. We develop a two-timescale input-driven Hopfield network in which fast associative retrieval is modulated by a slow feedback variable evolving over a prescribed memory-transition graph. This yields autonomous, graph-constrained transitions while preserving analytical tractability. Using slow-fast and geometric singular perturbation theory, we characterize memory fixation via memory manifolds and identify the geometric regions that govern stability loss and escape from retrieved memories. We further derive reduced slow dynamics and explicit criteria for transition onset, self-sustained retrieval, and collapse, expressed through escape times, gain thresholds, and fixed points. Together, these results provide a tractable framework for analyzing feedback-driven sequential retrieval in Hopfield networks.
ER -